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International Journal of Mathematics and Statistics Invention (IJMSI)
 

International Journal of Mathematics and Statistics Invention (IJMSI)

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International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI ...

International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.

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    International Journal of Mathematics and Statistics Invention (IJMSI) International Journal of Mathematics and Statistics Invention (IJMSI) Document Transcript

    • International Journal of Mathematics and Statistics Invention (IJMSI) E-ISSN: 2321 – 4767 || P-ISSN: 2321 - 4759 www.ijmsi.org || Volume 2 || Issue 2 || February - 2014 || PP-01-06 Optimal expressions for solution matrices of single – delay differential systems. C. Ukwu and E.J.D. Garba Department of Mathematics University of Jos, P.M.B 2084, Jos, Nigeria ABSTRACT: This paper derived optimal expressions for solution matrices of single – delay autonomous linear differential systems on any given interval of length equal to the delay h for non –negative time periods. The formulation and the development of the theorem exploited an earlier work Ukwu (2013b) on the interval [0, 4h]. The proofs were achieved using ingenious combinations of summation notations, multiple product notations and integrals, as well as the method of steps to obtain these matrices on successive intervals of length equal to the delay h. This theorem globally extends the time scope of applications of these matrices to the solutions of initial function problems, rank conditions for controllability and cores of targets, constructions of controllability Grammians and admissible controls for transfers of points associated with controllability problems. KEYWORDS- Delay, Matrices, Solution, Structure, Systems. I. INTRODUCTION The qualitative approach to the controllability of functional differential control systems have been areas of active research for the past fifty years among control theorists and applied mathematicians in general. This circumvents the severe difficulties associated with the search for and computations of solutions of such systems. Unfortunately computations of solutions cannot be wished away in the tracking of trajectories and many practical applications. Literature on state space approach to control studies is replete with variation of constants formulas, which incorporate the solution matrices of the free part of the systems. See Chukwu (1992), Gabsov and Kirillova (1976), Hale (1977), Manitius (1978), Tadmore (1984), and Ukwu (1987, 1992,1996 ). Regrettably no author has made any attempt to obtain general expressions for such solution matrices involving the delay h . The usual approach is to start from the interval [ 0 , h ] and compute the solution matrices and solutions for given problem instances and then use the method of steps to extend these to the intervals [ kh , ( k  1) h ], for positive integral k , not exceeding 2, for the most part. Such approach is rather restrictive and doomed to failure in terms of structure for arbitrary k . In other words such approach fails to address the issue of the structure and computing complexity of solution matrices and solutions quite vital for real-world applications. The need to address such short-comings has become imperative; this is the major contribution of this paper, with its wideranging implications for extensions to double-delay and neutral systems and holistic approach to controllability studies. II. PRELIMINARIES We consider the class of double-delay differential systems:  x ( t )  A 0 x ( t )  A1 x ( t  h ) , t  R (1) w h e r e A 0 a n d A 1 a r e n  n c o n s ta n t m a tr ic e s a n d x ( . ) L e t Y k  i ( t  ih ) be a solution is an n -dimensional column vector function. matrix J k  i  [ ( k  i ) h , ( k  1  i ) h ] , k  {0 , 1,  } , i  {0 , 1} , of (1), on the interval solution matrices, where; In ,t  0 Y (t )   0, t  0. Note: Y ( t ) is a generic solution matrix for any t R and I n is the identity matrix of order n . The solution matrices will be obtained piece – wise on successive intervals of length h . Ukwu (2013b) obtained the following expressions for the Y ( t ) o n J k , fo r k  {0 ,  , 3} : www.ijmsi.org 1|Page
    • A derivation of an optimal expression for solution matrices…   A0 t e , t  J0; t   e A0 t  e A0 ( t  s ) A e A0 ( s  h ) d s , t  J ; 1 1   h  t t s2  h  A0 t A0 ( t  s ) A0 ( s  h ) A (t  s2 ) A (s s h) A (s h ) Y (t )   e  e A1 e ds    e 0 A1 e 0 2 1 A1 e 0 1 d s 1 d s 2 , t  J 2 ;    e A0 t        h 2h t e A1 e A0 ( s  h ) h 2h t s3  h (4) s2  h   ds  (3) h t A0 ( t  s ) (2) e A0 ( t  s 2 ) A1 e A 0 ( s 2  s1  h ) A1 e A 0 ( s1  h ) d s1 d s 2 h s2  h    3h A0 ( t  s 3 ) A0 ( s 3  h  s 2 ) A 0 ( s 2  h  s1 ) A 0 ( s1  h ) h 2h e A1 e A1 e A1 e d s1 d s 2 d s , t  J 3 3 (5 ) He also interrogated some topological dispositions of the solution matrices and deduced that the solution matrices are continuous on the interval [0,4h] but not analytic t  {0 , h , 2 h , 3 h } . These results are consistent with the existing qualitative theory on Y ( t ). See Chukwu (1992), Hale (1977), Tadmore (1984) and Ukwu (1987, 1996). See also analytic function (2010) and Chidume (2007) for discussions on analytical functions and topology. The objective of this paper is to formulate and prove a theorem on the general  0 , 1,   , by appropriating the above expression for e x p r e s s io n f o r Y ( t ) o n J k , f o r k  Y ( t ). Theorem: Ukwu-Garba’s Solution Matrix Formula for Autonomous, Single – Delay Linear Systems: 1. e 0 , t  J0;  t  A0 t A (t  s ) A (sh ) e  e 0 A1 e 0 ds, t  J1;  h   t Y ( t )   A0 t A (t  s ) A (sh) e  e 0 A1 e 0 ds  h  si  h j   k t  A0 ( t  s j ) A ( s  h  s i 1 ) A 0 ( s1  h ) A1 e 0 i    e  A1 e   ds , t  J k , k  2    j  2 jh i  j , j  1 ,  , 2  ( i  1 ) h  1     A t (6 ) (7 ) (8 ) Proof The expressions (2) and (3) prove (6) and (7) respectively. If j  2 , in (8) we obtain: t Y (t )  e A0 t  s2  h t e A0 ( t  s ) A1 e A0 ( s  h ) ds  h e A0 ( t  s j )  2h  Y (t )  e  e t A0 ( t  s ) A1 e A0 ( s  h ) A0 ( s 2  h  s 1 ) A1 e A 0 ( s1  h ) d s1 d s 2 , t  J 2 h t A0 t A1 e ds  h s2  h   2h A0 ( t  s 2 ) e A1 e A0 ( s 2  h  s 1 ) A1 e A 0 ( s1  h ) d s1 d s 2 , t  J 2 (9 ) h The expression (9) agrees with (4); therefore the theorem is valid for t  J 2 . If j  3, in (8), we get: t Y (t )  e A0 t  e A1 e A0 ( s  h ) ds  h  3h e A0 ( t  s 3 )  2h e A0 ( t  s 2 ) 2h s3  h t  s2  h t A0 ( t  s )  A1 e A0 ( s 2  h  s 1 ) A1 e A0 ( s 1  h ) d s1 d s 2 2h s2  h A1 e A0 ( s 3  h  s 2 )  A1 e A0 ( s 2  h  s 1 ) A1 e A 0 ( s1  h ) d s1 d s 2 d s 3 h www.ijmsi.org 2|Page
    • A derivation of an optimal expression for solution matrices… t A0 t  Y (t )  e e  s2  h t A0 ( t  s ) A0 ( s  h ) A1 e   ds  h 2h s3  h t A0 ( t  s 2 ) e A1 e A0 ( s 2  h  s 1 ) A1 e A0 ( s 1  h ) d s1 d s 2 2h s2  h    3h A0 ( t  s 3 ) A0 ( s 3  h  s 2 ) A0 ( s 2  h  s 1 ) A 0 ( s1  h ) h  2h e A1 e A1 e A1 e d s1 d s 2 d s 3 (1 0 ) This result is consistent with the expression (5). Therefore the theorem is also valid for t  J 3 . The rest of the proof is inductive. Assume the validity of the theorem on J p , 4  p  k for some integer k . Then on J k  1 we have: t Y (t )  e A 0  t [ k 1]h  Y  [ k  1] h    A0  t  s k 1  e A1 Y  s k 1  h  d s k 1 (1 1) ( k 1) h s k 1  J k 1  s k 1  h  J k  Y ( s k 1  h ) a n d Y the induction hypothesis applies to [k  1] h  . Therefore:  k 1 h A0  k  1  h Y ([ k  1] h )  e   e A0 ( [ k  1 ] h  s 1 ) A0 ( s 1  h ) A1 e ds1 (1 2 ) h  k    j2   k 1 h si  h  e A0 ( [ k  1 ] h  s j )   A1 e A 0 ( s i  h  s i 1 ) i  j , j  1 ,  , 2  ( i  1 ) h jh j  A (s h )  A1 e 0 1  d s  , t  J k , k  2  1   (1 3 )  k 1 h A0 t  Y (t )  e   A0 ( t  s 1 ) e A1 e A0 ( s 1  h ) ds1 h  k    j2   k 1 h  si  h A0 ( t  s j ) e    e A1 e A0 ( s k 1  h )  d s k 1   k 1 h  k    j2  e A0 ( t  s k 1 )  e A0 ( t  s k 1 )  k 1 h e 1 A0 ( s k 1  h  s j )   k 1 h     k     j2 A0 ( t  s 1 ) A1 e A0 ( s 1  h )   k 1 h  e A0 ( t  s k 1 ) si  h e A0 ( t  s j )   A1 e A1  k 1 h A 0 ( s i  h  s i 1 ) i  j , j  1 ,  , 2  ( i  1 ) h jh A0 ( t  s k 1 ) A 0 ( s i  h  s i 1 ) A1 e A0 ( s k 1 d s1 d s k  1 (1 5 ) j  A (s h)  A1 e 0 1  d s  d s k  1 (1 6 )  1   h) d s k 1 (1 7 )  j  A (s h)  A1 e 0 1  d s   1   si  h 1 e A0 ( s k 1  h  s j )   i   j , j  1,  , 2  ( i  1 ) h jh A1 e A 0 ( s i  h  s i 1 ) (1 8 ) j  A 0 ( s1  h )  A1 e  d s  d s k 1  1   (1 9 ) s k 1  h t   ds1  s k 1  h e A0 ( s 1  h )  k 1 h t  A1 e t e h  k    j2  A1 e i   j , j  1,  , 2  ( i  1 ) h jh A0 t A0 ( s k 1  h  s 1 ) si  h  A1 A1 e h s k 1  h  Y (t )  e  A1  k 1 h t (1 4 ) s k 1  h t A0 ( t  s k 1 ) j  A (s h)  A1 e 0 1  d s   1   A0 ( s i  h  s i 1 ) i  j , j  1 ,  , 2  ( i  1 ) h jh t  A1 e e A0 ( t  s k 1 ) A1  k 1 h  A1 e A0 ( s k 1  h  s 1 ) A1 e A0 ( s 1  h ) d s1 d s k  1 (20) h t  Y (t )  e A0 t  e A0 ( t  s 1 ) A1 e A0 ( s 1  h ) ds1 h   k 1   j3   k 1 h  jh si  h e A0 ( t  s j )   A1 e A 0 ( s i  h  s i 1 ) i  j , j  1 ,  , 2  ( i  1 ) h www.ijmsi.org j  A (s h) A1 e 0 1  d s    1   ( 2 1) 3|Page
    • A derivation of an optimal expression for solution matrices…  k 1 h s2  h   e A0 ( t  s j )  A1 2h  k    j3  A0 ( s 2  h  s 1 ) A1 e A1 e A0 ( s 1  h ) s k 1  h t  A0 ( t  s k 1 ) e d s1 d s 2 (22) h  A1  k 1 h  si  h e 1 A 0 ( s k 1  h  s j 1 )  A 0 ( s i  h  s i 1 ) i   j  1,  , 2  ( i  1 ) h j 1 h j 1  A (s h)  A1 e 0 1  d s  d s k  1  1   (2 3) s k 1  h t    A1 e e A0 ( t  s k 1 )  A1  k 1 h A0 ( s k 1  h  s 1 ) A1 e A1 e A0 ( s 1  h ) d s1 d s k  1 (24) h Note the use of change of variables in obtaining (23) and that j = k + 1 zeros out (21). (22) and (24) add up to yield: s2  h t e A0 ( t  s 2 )  A1 2h A0 ( s 2  h  s 1 ) A1 e A1 e A0 ( s 1  h ) d s1 d s 2 (25) h (23) may be rewritten in the form:  k   j3  si  h t  e A0 ( t  s k 1 )   A1 e A 0 ( s i  h  s i 1 ) i   j , j  1,  , 2  ( i  1 ) h  k 1 h j  A (s h)  A1 e 0 1  d s   1   (26) Hence (21) and (26) add up to yield:  k 1   j3  si  h t  e A0 ( t  s j )   A1 e A 0 ( s i  h  s i 1 ) i  j , j  1 ,  , 2  ( i  1 ) h jh j  A 0 ( s1  h )  A1 e  d s  1   (27) t Add up e A0 t  e A0 ( t  s 1 ) A1 e A0 ( s 1  h ) d s 1 ,(25) and (27) to get: h t Y (t )  A 0t e  e A0 ( t  s 1 ) A1 e A0 ( s 1  h ) ds1 h   k 1   j2  si  h t  e A0 ( t  s j ) jh   A1 e A 0 ( s i  h  s i 1 ) i  j , j  1 ,  , 2  ( i  1 ) h j  A 0 ( s1  h )  A1 e  d s , o n J k .  1   (28) So, the theorem is valid on J k  1 , and hence valid for all J k , k  { 0 , 1, 2 ,  } . This completes the proof. 3.1 Corollary 1 If A 1  d i a g ( b ) , t h e n : e 0 , t  J 0;  i i k Y (t )   b ( t  ih ) A 0 ( t  ih ) A0 t   e , t  Jk,k  1 e i! i 1  A t Proof The proof follows by straight-forward successive integration, noting that with A e 0 (29) (3 0 ) A 1  d ia g ( b )  A 1 commutes (.) ; S o Y ( t ) r e d u c e s to : www.ijmsi.org 4|Page
    • A derivation of an optimal expression for solution matrices… e 0 , t  J 0;  t  A0 t A (t  h ) A t A (t h ) e   be 0 d s  e 0  b (t  h )e 0 , t  J1;  h  Y (t )   A t A (t  h ) e 0  b (t  h )e 0  si  h   k t j  A0 ( s 1  h ) A0 ( t  s j ) A 0 ( s i  h  s i 1 )    be  e   d s , t  J k , k  2  be   j  2 jh i  j , j  1 ,  , 2  ( i  1 ) h  1     A t ( 3 1) (3 2 ) (3 3) The sum of the multiple integrals in (33) yields: t k  j b e A 0 ( t  jh )  i  j , j  1 ,  , 2  j2 Adding the expression e k Y (t )  e A0 t   b A 0t si  h   jh ( i 1) h i e i! i 1 d s 1d s 2  d s j  A 0 ( t  ih )  j b e A 0 ( t  jh ) ( t  jh ) j2  b (t  h )e ( t  ih ) i k A 0 (t  h ) j! j k A 0 ( t  ih )  b e i ( t  ih ) i (3 4 ) i! i 2 to the expression (3.39) yields the result: , t  Jk,k  2 (3 5 ) The expressions (31), (32) and (35) complete the proof of the corollary. 3.2 Corollary 2 If A 0  0 , th e n : In, t  J0;  i i k Y (t )   A 1 ( t  ih ) , t  Jk,k  1 In   i! i 1  Proof The proof follows by straight-forward successive integration, noting that with e A 0 (.) A0  0  e A  I n ; S o Y ( t ) re d u c e s to the result obtained by replacing b b y A 1 a n d e 0 A (.) 0  In  A1 (.) by I n commutes in corollary 1. The desired result is precisely that stated in the conclusion of the corollary. 3.3 Corollary 3 If n  1, A 0  a , A 1  b , th e n : e , t  J 0;  i i k Y (t )   b ( t  ih ) a ( t  ih ) at e   e , t  Jk,k  1  i! i 1  at Proof Proof follows immediately by replacing A 0 b y a a n d d i a g ( b ) b y b in corollary 1. 3.4 Remarks: Please note in particular that this result is consistent with the theorem in Ukwu and Garba (2013c). However it must be pointed out that it was that theorem that motivated our theorem and hence corollary 3. III. CONCLUSION This article has completely determined the structure solution matrices which are indispensible for the determination of all solutions of single-delay autonomous differential and control systems. Moreover the ingenious combinations of summation notations, multiple product notations, multiple integrals and change of variable technique are unprecedented in the achievement of the desired proofs. The ideas exposed in this paper can be exploited to extend the results to double-delay and neutral systems. www.ijmsi.org 5|Page
    • A derivation of an optimal expression for solution matrices… REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] Analytic function. (2010). Retrieved September 11, 2010 from http://en.wikipedia.org/wiki/Analytic_function. Chidume, C. (2007). Applicable Functional Analysis. The Abdus Salam, International Centre for Theoretical Physics, Trieste, Italy. Chukwu, E. N. (1992). Stability and Time-optimal control of hereditary systems. Academic Press, New York. Driver, R. D. (1977). Ordinary and Delay Differential Equations. Applied Mathematical Sciences 20, Springer-Verlag, New York. Gabasov, R. and Kirillova, F. (1976). The qualitative theory of optimal processes. Marcel Dekker Inc., New York. Hale, J. K. (1977). Theory of functional differential equations. Applied Mathematical Science, Vol. 3, Springer-Verlag, New York. Manitius, A. (1978). Control Theory and Topics in Functional Analysis. Vol. 3, Int. Atomic Energy Agency, Vienna. Tadmore, G. (1984). Functional differential equations of retarded and neutral types: Analytical solutions and piecewise continuous controls. J. Differential equations, Vol. 51, No. 2, Pp. 151-181. Ukwu, C. (1987). Compactness of cores of targets for linear delay systems., J. Math. Analy. and Appl., Vol. 125, No. 2, August 1. pp. 323-330. Ukwu, C. (1992). Euclidean Controllability and Cores of Euclidean Targets for Differential difference systems Master of Science Thesis in Applied Math. with O.R. (Unpublished), North Carolina State University, Raleigh, N. C. U.S.A. Ukwu, C. (1996). An exposition on Cores and Controllability of differential difference systems, ABACUS Vol. 24, No. 2, pp. 276-285. Ukwu, C. (2013a). On Determining Matrices, Controllability and Cores of Targets of Certain Classes of Autonomous Functional Differential Systems with Software Development and Implementation. Doctor of Philosophy Thesis, UNIJOS (In progress). Ukwu, C. (2013b).The structure of Solution Matrices for Certain Single Delay Systems, controllability exposition and application instances. (Manuscript for publication). Ukwu, C. and Garba, E. J. D. (2013c). Derivation of an Optimal Expression for Solution Matrices of a Class of Single – Delay Scalar Differential Equations. (Submitted for publication). www.ijmsi.org 6|Page